Two scenarios on a potential smoothness breakdown for the three-dimensional Navier-Stokes equations

In this paper we construct two families of initial data being arbitrarily large under any scaling-invariant norm for which their corresponding weak solution to the three-dimensional Navier-Stokes equations become smooth on either $[0,T_1]$ or $[T_2,\infty)$, respectively, where $T_1$ and $T_2$ are two times prescribed previously. In particular, $T_1$ can be arbitrarily large and $T_2$ can be arbitrarily small. Therefore, possible formation of singularities would occur after a very long or short evolution time, respectively. We further prove that if a large part of the kinetic energy is consumed prior to the first (possible) blow-up time, then the global-in-time smoothness of the solutions follows for the two families of initial data.Comment: 16 pages, no figures. Some typos have been correcte

Convergence to suitable weak solutions for a finite element approximation of the Navier-Stokes equations with numerical subgrid scale modeling

In this work we prove that weak solutions constructed by a variational multiscale method are suitable in the sense of Scheffer. In order to prove this result, we consider a subgrid model that enforces orthogonality between subgrid and finite element components. Further, the subgrid component must be tracked in time. Since this type of schemes introduce pressure stabilization, we have proved the result for equal-order velocity and pressure finite element spaces that do not satisfy a discrete inf-sup condition.Comment: 23 pages, no figure

Introducción al Método de Elementos Finitos.

Se presentará una introducción a la formulación variacional de problemas elípiticos lineales y a su resolución numérica mediante métodos de elementos finitos. Se discutirá la implementación del método y se presentarán sus propiedades de convergencia, estabilidad y orden. Se introducirá finalmente la extensión del método a problemas no lineales y de evolución.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

On the approximation of turbulent fluid flows by the Navier-Stokes-α\alpha equations on bounded domains

The Navier-Stokes-$\alpha$ equations belong to the family of LES (Large Eddy Simulation) models whose fundamental idea is to capture the influence of the small scales on the large ones without computing all the whole range present in the flow. The constant $\alpha$ is a regime flow parameter that has the dimension of the smallest scale being resolvable by the model. Hence, when $\alpha=0$, one recovers the classical Navier-Stokes equations for a flow of viscous, incompressible, Newtonian fluids. Furthermore, the Navier-Stokes-$\alpha$ equations can also be interpreted as a regularization of the Navier-Stokes equations, where $\alpha$ stands for the regularization parameter. In this paper we first present the Navier-Stokes-$\alpha$ equations on bounded domains with no-slip boundary conditions by means of the Leray regularization using the Helmholtz operator. Then we study the problem of relating the behavior of the Galerkin approximations for the Navier-Stokes-$\alpha$ equations to that of the solutions of the Navier-Stokes equations on bounded domains with no-slip boundary conditions. The Galerkin method is undertaken by using the eigenfunctions associated with the Stokes operator. We will derive local- and global-in-time error estimates measured in terms of the regime parameter $\alpha$ and the eigenvalues. In particular, in order to obtain global-in-time error estimates, we will work with the concept of stability for solutions of the Navier-Stokes equations in terms of the $L^2$ norm

Convergence to suitable weak solutions for a finite element approximation of the Navier–Stokes equations with numerical subgrid scale modeling

In this work we prove that weak solutions constructed by a variational multiscale method are suitable in the sense of Scheffer. In order to prove this result, we consider a subgrid model that enforces orthogonality between subgrid and finite element components. Further, the subgrid component must be tracked in time. Since this type of schemes introduce pressure stabilization, we have proved the result for equal-order velocity and pressure finite element spaces that do not satisfy a discrete inf-sup conditio

Inf-Sup Stable Finite Element Methods for the Landau--Lifshitz--Gilbert and Harmonic Map Heat Flow Equations

In this paper we propose and analyze a finite element method for both the harmonic map heat and Landau–Lifshitz–Gilbert equation, the time variable remaining continuous. Our starting point is to set out a unified saddle point approach for both problems in order to impose the unit sphere constraint at the nodes since the only polynomial function satisfying the unit sphere constraint everywhere are constants. A proper inf-sup condition is proved for the Lagrange multiplier leading to the well-posedness of the unified formulation. A priori energy estimates are shown for the proposed method. When time integrations are combined with the saddle point finite element approximation some extra elaborations are required in order to ensure both a priori energy estimates for the director or magnetization vector depending on the model and an inf-sup condition for the Lagrange multiplier. This is due to the fact that the unit length at the nodes is not satisfied in general when a time integration is performed. We will carry out a linear Euler time-stepping method and a non-linear Crank–Nicolson method. The latter is solved by using the former as a non-linear solver.Ministerio de Economía y Competitividad MTM2015-69875-

On the approximation of turbulent fluid flows by the Navier-Stokes-α equations on bounded domains

The Navier-Stokes- equations belong to the family of LES (Large Eddy Simulation) models whose fundamental idea is to capture the influence of the small scales on the large ones without computing all the whole range present in the flow. The constant is a regime flow parameter that has the dimension of the smallest scale being resolvable by the model. Hence, when = 0, one recovers the classical Navier-Stokes equations for a flow of viscous, incompressible, Newtonian fluids. Furthermore, the Navier-Stokes- equations can also be interpreted as a regularization of the Navier- Stokes equations, where stands for the regularization parameter. In this paper we first present the Navier-Stokes- equations on bounded domains with no-slip boundary conditions by means of the Leray regularization using the Helmholtz operator. Then we study the problem of relating the behavior of the Galerkin approximations for the Navier-Stokes- equations to that of the solutions of the Navier-Stokes equations on bounded domains with no-slip boundary conditions. The Galerkin method is undertaken by using the eigenfunctions associated with the Stokes operator. We will derive local- and global-in-time error estimates measured in terms of the regime parameter and the eigenvalues. In particular, in order to obtain global-in-time error estimates, we will work with the concept of stability for solutions of the Navier-Stokes equations in terms of the L2 norm.Ministerio de Educación y Ciencia JC2011-041

Stability and convergence of two discrete schemes for a degenerate solutal non-isothermal phase-field model

We analyze two numerical schemes of Euler type in time and C0 finite-element type with P1-approximation in space for solving a phase-field model of a binary alloy with thermal properties. This model is written as a highly non-linear parabolic system with three unknowns: phase-field, solute concentration and temperature, where the diffusion for the temperature and solute concentration may degenerate. The first scheme is nonlinear, unconditionally stable and convergent. The other scheme is linear but conditionally stable and convergent. A maximum principle is avoided in both schemes, using a truncation operator on the L2 projection onto the P0 finite element for the discrete concentration. In addition, for the model when the heat conductivity and solute diffusion coefficients are constants, optimal error estimates for both schemes are shown based on stability estimates.Ministerio de Educación y CienciaCGCI MECD-DGU Brazil/Spai

Conditional stability and convergence of a fully discrete scheme for three-dimensional Navier–Stokes equations with mass diffusion

We construct a fully discrete numerical scheme for three-dimensional incompressible fluids with mass diffusion (in density-velocity-pressure formulation), also called the Kazhikhov–Smagulov model. We will prove conditional stability and convergence, by using at most C0-finite elements, although the density of the limit problem will have H2-regularity. The key idea of our argument is first to obtain pointwise estimates for the discrete density by imposing the constraint lim(h,k)→0 h/k = 0 on the time and space parameters (k, h). Afterwards, under the same constraint on the parameters, strong estimates for the discrete density in l ∞(H1) and for the discrete Laplacian of the density in l 2(L2) are obtained. From here, the compactness and convergence of the scheme can be concluded with similar arguments as we used in [Math. Comp., to appear], where a different scheme is studied for two-dimensional domains which is unconditionally stable and convergent. Moreover, we study the asymptotic behavior of the numerical scheme as the diffusion parameter λ goes to zero, obtaining convergence as (k, h, λ) → 0 towards a weak solution of the density-dependent Navier–Stokes system provided that the constraint lim(λ,h,k)→0 h/(λ2k) = 0 on (h, k, λ) is satisfied.Ministerio de Educación y Cienci